Gemara

Version #1 (Beraisa): "Tizbach" teaches that (you may slaughter Pesulei ha'Mukdashim, but) you may not shear them. "(You may eat the) Vasar" (meat, but) not the milk. "V'Achalta" -- (you may eat it), but not for your dogs;

This is the source that we do not redeem Kodshim in order to feed them to dogs.

Version #2 (Beraisa): "Tizbach v'Achalta" - you may eat only after Shechitah.

This version permits redeeming Kodshim in order to feed them to dogs.

32b (Mishnah - Beis Shamai): A Kohen may not invite a Yisrael to eat Bechor (a Ba'al Mum);

Beis Hillel permit inviting even a Nochri.

(Gemara): Our Mishnah is like R. Akiva;

(Beraisa - Beis Shamai): Only Kohanim may be invited to eat Bechor;

Beis Hillel permit inviting Yisraelim;

R. Akiva permits even Nochrim.

Beis Shamai learn from "u'Vesaram Yihyeh Lach (ka'Chazeh ha'Tenufah uch'Shok... )" - just like Chazeh v'Shok (the chest and thigh of Shelamim, which are Matnos Kehunah) are permitted to Kohanim and forbidden to Yisrael, also Bechor.

Beis Hillel hold that this applies only to a Tam Bechor. A Ba'al Mum may be eaten "ka'Tzvi vecha'Ayal" (Chayos, which have no Kedushah);

A Tamei (Kohen) may not eat Kodshei Kalim, yet he may eat Bechor (Ba'al Mum). A (Tahor) Zar may eat Kodshei Kalim, all the more so he may eat Bechor!

R. Akiva learns from "ka'Tzvi vecha'Ayal" - just like deer and wild goats are permitted to Nochrim, also a Bechor Ba'al Mum.

Question: Why do Beis Hillel (in the Beraisa) argue?

Answer: Three verses mention Tzvi va'Ayal;

One of them teaches the laws of R. Yitzchak and R. Oshaya. (One who works with Pesulei ha'Mukdashim after it was redeemed or mates it is lashed);

One of them teaches like R. Elazar Hakapar expounded (that Chayos must be slaughtered, just like animals);

The third teaches that just like Bechorah does not apply to Tzvi va'Ayal, it does not apply to Pesulei ha'Mukdashim.

Rishonim

Rambam (Hilchos Bechoros 1:3): A Kohen may eat a blemished Bechor anywhere. He may sell it or feed it to whoever he wants, even to a Nochri, for it is Chulin.

Rosh (5:2): We follow Beis Hillel in the Mishnah, unlike Beis Hillel in the Beraisa. When there is a Stam Mishnah and an argument in a Beraisa, the Halachah follows the Mishnah (Yevamos 42b). All the more so we say so here, for Beis Hillel is better than a Stam Mishnah (Shabbos 157a). Beis Hillel of the Beraisa expound ka'Tzvi vecha'Ayal like R. Elazar Hakapar, to teach that Pesulei ha'Mukdashim require Shechitah. He holds that the Torah does not require Shechitah for birds. Most Amora'im hold like R. Akiva, who says that the Torah requires Shechitah for birds. Therefore, if a Bechor died or was slaughtered without a Heter (from a Chacham who inspected the Mum) or it was found to be Terefah, it must be buried, even according to R. Akiva. Bahag connotes that he rules like Beis Hillel of the Beraisa. In Pesachim (29a), we expound "v'Achalta", and not for your dog, and this excludes also Nochrim. Do not learn from here that the Bechor is forbidden to Nochrim. Even though several Stam Mishnayos say that a Bechor that died is Asur b'Hana'ah, this is only when Yisre'elim may not eat the meat. When it is permitted to Yisrael, we learn from Tzvi va'Ayal that it is permitted also to Nochrim. We forbid due to v'Achalta only when it is forbidden to Yisrael.

Ran (Chulin 25a DH Yatza): A Mishnah (Chulin 69b) teaches that if the majority of a Bechor left the womb (intact, and then it died), it must be buried. This implies that that one may not sell it to a Nochri, for it is Asur b'Hana'ah. Zevachim 103b connotes similarly. It says that if a Bechor was found to be Terefah, Kohanim may benefit from the skin. This implies that the meat is Asur b'Hana'ah, and it must be buried. However, Bechoros 32b connotes that one may benefit from it, and invite even a Nochri to eat from it. Tosfos (Chulin 69b DH Yatza) answered that whenever one may eat it with a Mum, one may invite a Nochri to eat from it, for the Torah equates it to Tzvi va'Ayal. However, one may not feed it to dogs, for it says "v'Achalta", but not your dog. When one may not eat it, e.g. it is Terefah, there is no Hekesh, so it is Asur b'Hana'ah. We find that R. Akiva permits Hana'ah (only) from the skin, yet he holds that one may invite a Nochri to eat from it! We must answer like I said.

Mordechai (Yevamos Sof Siman 126): One may feed a Bechor Ba'al Mum to a Nochri or dog. If it was found to be Terefah, one may not sell it to a Nochri or feed it to a dog. One must bury the meat. Therefore, when it is blemished, it is proper to sell it to Nochrim while it is alive.

Poskim

Shulchan Aruch (YD 306:5): Nowadays, we slaughter a Bechor after it gets a Mum and eat it (anywhere). One may feed it even to a Nochri (or dogs).

Beis Yosef (DH u'Mah she'Chasav u'Ma'achilo): The Rambam and Rosh rules like R. Akiva. Bahag rules like Beis Hillel in the Beraisa.

Shach (6): If it is Terefah one may not benefit from it, like we say in 307:2.

Yam Shel Shlomo (Chulin 4:4): The Ran expounds from "v'Achalta" that one may not feed it to dogs. This is wrong. There is no more Heter to a Nochri than to your dog, for you must feed your dog. The exclusion for dogs applies only to a Terefah, like the Mordechai says explicitly. Also the Rosh connotes like this.

Shach (7): The Maharshal says that mid'Rabanan, one may not sell it to Nochrim. One may feed a Nochri Bechor only if he invites him to eat with the Yisrael. He may not sell to him, or even give him a raw piece. Likewise, one may feed to dogs only what remains from the meat with the bones, but not to overtly cast it to dogs.

Shach (8): The Maharshal did not give a proper reason to argue with the Ran (who forbids giving to dogs). There is no connotation that the Rosh permits. The Maharshal infers from the Seifa, in which the Rosh forbids a Terefah to dogs and Nochrim, that in the Reisha (a Kosher Ba'al Mum), he permits to both of them. This is wrong. Rather, in the Seifa he forbids to dogs, and the same applies to Nochrim, which we learn from dogs. Also Tosfos (32b DH Posak) permits to Nochrim; we do not forbid due to "and not to your dogs." This implies that it is forbidden to dogs. Also the Rashba (Chulin 69b DH Masnisin) says that when Yisre'elim may eat it, even Nochrim may eat it, but not dogs, for it says v'Achalta. Also R. Yerucham (20:2) says so. Therefore, one should be careful (not to feed it to dogs).

Shulchan Aruch (307:2): If it was found to be Terefah, one may not benefit from it meat or skin. One must bury it. Therefore it is good to sell it to a Nochri before (Shechitah).)

Beis Yosef (DH v'Im): Rashi (Zevachim 104a) explains that the meat is buried because we do not feed Kodshim to dogs, and we burn Kodshim only if they entered the Azarah and became Pasul after Shechitah.

Beis Yosef (DH Matzasi): I found in a Ashkenazi Teshuvah (Maharil 42) in the name of Sefer ha'Parnes (from a Talmid of the Maharam me'Rotenburg, Siman 304) that if a Bechor has a Mum, it is good to sell it to a Nochri while it is alive.

Yam Shel Shlomo (DH Bechor): If a Chachamim permitted a Bechor Ba'al Mum, and one fears lest it become Terefah and he will need to bury it, I say that one may not sell it to a Nochri. The Beis Yosef brought from ha'Parnes that it is proper to sell it to Nochrim while it is alive, but I am not concerned for this. If the lungs were lost, we consider it to be forbidden, since Terefos of the lings are common. We say that after Shechitah it has Chezkas Heter only in cases such as a wolf that took the intestines and returned them punctured. Rashi is lenient, but the Ge'onim disagree. Therefore, I say that it has Chezkas Isur, and one may not sell it to Nochrim. However, we forbid a Safek (Terefah) only if it is a Vadai Bechor. If it is a Safek Bechor, we are lenient due to a Sefek-Sefeka (two doubts). However, I hold that even a Vadai Kosher Ba'al Mum should not be sold to Nochrim, for this is a disgrace to Kodshim. This is worse than selling it in the market. Beis Hillel permit only inviting a Nochri to eat from it, but not to selling to Nochrim or even giving a piece to Nochrim, or casting it to dogs. It is not a proper Isur like the Ran thought. "Not to your dogs" merely excludes a Terefah. The Mordechai permits giving to dogs, i.e. what is left in the bowl with the bones, but not to overtly feed to dogs.

Rebuttal (Shach 1): Most Poskim permit when the lungs were lost. Even those who forbid, this is only because if not, Chachamim's decree did not help (people will intentionally discard the lungs without checking them). This does not apply here (selling a Bechor to a Nochri). Also Maharil (72) and the Mordechai permit selling to a Nochri. However, Maharil brings that the Terumas ha'Deshen forbids selling to a Nochri, and also Maharam of Tikatin in his comments on the Mordechai forbids in the name of Teshuvas Acharonim of the Or Zaru'a.

R. Akiva Eiger: Chacham Tzvi (2) forbids selling to a Nochri, lest the Nochri sell it to a Yisrael, and the Yisrael will shear it or work with it. He forbids even selling on condition that the Nochri slaughter it in front of us.

Pischei Teshuvah (1): The Chasam Sofer (303) forbids selling it to a Nochri, for surely he will make it a Nevelah, and it must be buried. However, a Yisrael may slaughter it in a Kosher way and sell it to a Nochri without checking the lungs.